
where \(\epsilon _t\)  specifies the step size. The initial distribution \(\mathbf {x}_T\)  is sampled from a given prior distribution and the noise \(\mathbf {z}_t\)  are i.i.d. samples of the standard \(d\) -dimensional Gaussian distribution.
Under some regularity conditions, if \(\epsilon _t\rightarrow 0,T\rightarrow +\infty \) , we have the terminal distribution \(p(\mathbf {x}_0)\)  that approaches \(p_{\mathbf {x}^{\ast }}\)  [1]}.
With this process, we transform a sample drawn from an initial Gaussian distribution to approach a desired distribution \(p_{\mathbf {x}^{\ast }}\) .
